Question

A consulting firm had predicted that

29%

of the employees at a large firm would take advantage of a new company Credit​ Union, but management is skeptical. They doubt the rate is that high. A survey of

200

employees shows that

82

of them are currently taking advantage of the Credit Union.

​a) Find the standard deviation of the sample proportion based on the null hypothesis.

​b) Find the​ z-statistic.

​c) Does the​ z-statistic seem like a particularly large or small​ value?

Answer 1

Given that, 29% of the employees at a large firm would take advantage of a new company Credit Union.

That is,

Population Proportion of employees who take advantage = P = 0.29

Now, a sample of 200 employees are selected and 82 of them are currently taking advantage of the Credit Union.

Here, each employee either take advantage or not taking advantage. there are only two possibilities for every employee.

Let, X = number of employees out of 200 employees taking advantage of the Credit Union.

Therefore, X follows a Binomial distribution with parameter n = 200 and probability of success, P = 0.29.

Now, sample proportion is denoted by p and is given by

p = X/n = 82/200 = 0.41

Here, we have to test the hypothesis

H₀: P = 0.29

H₁: P 0.29

Now, the test statistic for testing the above hypothesis is,

Now,

E(X) = n*P = 200*0.29 = 58

and Variance(X) = V(X) = n*P*(1-P) = 200*0.29*0.71 = 41.18

Now, variance of the sample proportion is

Now under the null hypothesis, P = P₀ = 0.29. Therefore

Hence, the standard deviation of the sample proportion (SD(p)) based on the null hypothesis is

Therefore, the standard deviation of the sample proportion based on the null hypothesis is 0.0321.

b)

The test statistic Z is

Therefore the Z statistic is 3.74.

c)

Let the level of significance = =0.05

Therefore the critical value is

Since the value of test statistic = 3.74 > critical value = 1.96, therefore at 5% level of significance, reject the null hypothesis and conclude that at 5% level of significance, the proportion of employees taking advantage is not equal to 0.29.

Also, the Z-statistic seem like a particularly large or small value.